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Frege's Influence on Wittgenstein:Reversing Metaphysics via the Context Principle*
Erich H. Reck
Gottlob Frege and Ludwig Wittgenstein (the later Wittgenstein) are often seen as polaropposites with respect to their fundamental philosophical outlooks: Frege as a paradig-matic realist , Wittgenstein as a paradigmatic anti-realist . This opposition is supposedto find its clearest expression with respect to mathematics: Frege is seen as the arch-pla-tonist , Wittgenstein as some sort of radical anti-platonist . Furthermore, seeing them assuch fits nicely with a widely shared view about their relation: the later Wittgenstein issupposed to have developed his ideas in direct opposition to Frege. The purpose of thispaper is to challenge these standard assumptions. I will argue that Frege's and Wittgen-stein's basic outlooks have something crucial
in common
; and I will argue that this is theresult of the positive
influence
Frege had on Wittgenstein.It would be absurd to claim that there are no important differences between Frege andWittgenstein. Likewise, it would be absurd to claim that the later Wittgenstein was notcritical of some of Frege's ideas. What, then, is the common element I see? My sugges-tion is that the two thinkers agree on what I call a
reversal of metaphysics
(relative to astandard kind of metaphysics attacked by both). This is not an agreement on one particu-lar thesis or on one argument. Rather, it has to do with what is
prior
and what is
posterior
when it comes to certain fundamental explanations in metaphysics (also,relatedly, in semantics and epistemology). Furthermore, this explanatory reversal is inti-mately connected with Frege's
context principle
: Only in the context of a sentence dowords have meaning . As we will see, Wittgenstein takes over this principle andmodifies it to: Only in the practice of a language can a word have meaning .The context principle has not gone unnoticed in the literature, indeed it has received anumber of different interpretations. However, none of them does justice to Frege's andWittgenstein's understanding of it; and this is, it seems to me, directly connected with thestandard ways of categorizing Frege and Wittgenstein: as realist and anti-realist.
*
This paper is a shortened version of (Reck 1997). That paper, in turn, grew out of my dissertation
Frege, Wittgenstein, and Platonism in Mathematics
(University of Chicago, 1992), written under the super-vision of W. W. Tait, Leonard Linsky, and Michael Forster. I would, once more, like to thank thefollowing people for their help with the srcinal paper: Steve Awodey, Andre Carus, Stuart Glennan, RobinJeshion, Christoph Lankers, Michael P. Price, and Gisbert W. Selke. In addition, I am grateful to MichaelBeaney for suggestions on how to shorten the paper.
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According to the reading to be developed here, the context principle (in Frege
and
Witt-genstein)
cuts across
the usual realism-vs.-antirealism distinction. What is more, thereversal of metaphysical explanations with which the principle is tied up undermines thevery
basis
for this distinction. Now, this very fact makes it also harder to explain mynew perspective on Frege and Wittgenstein—since it is at odds with certain widespread(though often only implicit) assumptions in contemporary metaphysics, philosophy of language, and philosophy of mathematics. Some of these assumptions will, then, alsohave to be subjected to critical scrutiny in the course of this paper.At the same time, my approach may at first glance not appear to diverge very far fromthe mainstream, particularly with respect to Frege. For instance, it will turn out that myFrege is a
platonist
, that is to say: he does view mathematical judgments as objective; hedoes hold that in the corresponding statements our number terms refer to numbers; and hedoes maintain that these numbers are non-mental, non-physical, self-subsistent objects.In other words, I do not want to deny at all that Frege makes such statements, not eventhat he means them seriously. Instead, I want to re-interpret
what he means
by them
, i.e.,
what kind
of platonist
he is. I will argue that Frege is a
contextual
platonist
, not a
meta- physical
platonist
—a crucial difference which should be clear by the end of the paper. Inaddition, if my new reading of Frege is correct, it becomes possible to see that his platon-ism is less in opposition to Wittgenstein's anti-platonism than is usually assumed. I willsubstantiate this conclusion by re-considering Wittgenstein's position, too, with respect toboth his critical and his constructive views.
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I. FREGE AND METAPHYSICAL PLATONISM
Throughout his life Frege's main goal was to put arithmetic on a firm foundation. Forhim this amounted to analyzing and clarifying its logical structure, thus revealing bothwhat it is based on and what it is about. In other words, Frege's logical investigationslead him to certain epistemological and metaphysical conclusions: about the basis of arithmetic judgments and the nature of numbers (and functions). As is well-known,Frege's conclusions amount to a kind of
logicism
; that is, for him arithmetic judgmentsfind their foundation in basic logical laws and numbers turn out to be logical entities.This much of the standard reading I do not want to question. More problematic for me—
1
With respect to my whole perspective (on Frege, Wittgenstein, and platonism) I have been guided by(Tait 1986a). My interpretation of Frege has, in addition, been influenced by (Dummett 1978a), (Ricketts1986), and the first few chapters of (Diamond 1991). Similarly, my interpretation of Wittgenstein has alsobeen influenced by (Tait 1986b) and by Steve Gerrard's dissertation
Wittgenstein in Transition. The Philos-ophy of Mathematics
(University of Chicago, 1986), published in part as (Gerrard 1991).
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i.e., more in need of clarification—is that Frege's position is usually also characterized asa kind of
platonism
.
Frege's Platonism (Vaguely and Naively).
What exactly it means to say that Frege is a platonist —or what could be meant by platonism in the first place—is a central ques-tion in this paper. I do not think there is only one possible answer. But let us first look atsome
typical characterizations
of platonism . For instance, in the
Encyclopedia of Philosophy
we read:
By platonism is understood the realistic view, akin to that of Plato himself, that abstractentities exist in their own right, independently of human thinking. According to this viewnumber theory is to be regarded as the description of a realm of objective, self-subsistentmathematical objects that are timeless, non-spatial, and non-mental. Platonism conceivesit to be the task of the mathematician to explore this and other realms of being. Amongmodern philosophers of mathematics Frege is a pre-eminent representative of platonism,distinguished by his penetrating lucidity and his intransigence. (p. 529)
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This passage contains the main elements of the understanding of platonism dominant inmuch of recent metaphysics, philosophy of language, and philosophy of mathematics. Itfocuses on the following three claims: (i) numbers and other mathematical entities are abstract objects which exist in their own right ; (ii) in mathematics we describe these objects, i.e., we talk about them as members of a mathematical realm ; and (iii) thetask of the mathematician is to explore this realm, i.e., to find out what is objectivelythe case in it. At the same time, an understanding of platonism which just cites thesethree claims (without further explication) is still very vague and naive. In fact, it will turnout to be
ambiguous
, i.e., allow for two rather different interpretations.It is hard to deny, though, that Frege is a platonist in this vague and naive sense. Heis most explicit about his views in this connection in
Foundations of Arithmetic
(1884).There he says about numbers as objects:
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[S]urely the number one looks like a definite object, with properties that can be specified,for example that of remaining unchanged when multiplied by itself. (FA, p. II)But it will perhaps be objected, even if the earth is really not imaginable, it is at any ratean external thing, occupying a definite place; but where is the number 4? It is neitheroutside us nor within us. And, taking those words in their spatial sense, that is quitecorrect. ... Yet the only conclusion to be drawn from that is that 4 is not a spatial object,not that it is not an object at all. Not every object has a place. (
Ibid
, p. 72)
And about the nature of numbers and the objectivity of arithmetic:
For number is no whit more an object of psychology or a product of mental processesthan, let us say, the North Sea. (
Ibid
, p. 34)
2
See (Barker 1967).
3
I will quote from Frege's writings using the following (standard) abbreviations: BL for
The Basic Laws of Arithmetic
(Frege 1967); CP for
Collected Papers
(1984); FA for
Foundations of Arithmetic
(1968); and PW for
Posthumous Writings
(1979). Notice, however, that in a number of cases I havefound it necessary to
amend
the usual translations
.
- 4 -But arithmetic is no more psychological than, say, astronomy. Astronomy is concerned,not with ideas of the planets, but with the planets themselves, and by the same token theobjects of arithmetic are not ideas either. (
Ibid
, p. 37)Even the mathematician cannot create things at will, any more than the geographer can;he too can only discover what is there and give it a name. (
Ibid
, pp. 107-8)
Clearly all the three main ingredients of platonism mentioned above are contained inthese remarks. (Note, however, that Frege does not use the term 'abstract object' fornumbers; he prefers 'logical object', for reasons which will become clear later.)Another work often cited in connection with Frege's platonism is his late article Thoughts (1918-19). As the title suggests, in this article Frege is mostly concernedwith the nature of thoughts , not with numbers. Thoughts in his sense are the contentsof judgments—they are what can be asserted, believed, questioned, etc. And they, too,turn out to exist as non-mental and non-physical objects; or as Frege puts it now, theyexist in a special intellectual realm (which also contains numbers). Thus he writes:
A third realm must be recognized. Anything belonging to this realm has it in commonwith ideas that it cannot be perceived by the senses, but has it in common with things thatit does not need an owner so as to belong to the contents of his consciousness. Thus forexample the thought we have expressed in the Pythagorean theorem is timelessly true,true independently of whether anyone takes it to be true. It needs no owner. It is not trueonly from the time when it is discovered; just as a planet, even before anyone saw it, wasin interaction with other planets. (CP, p. 363)
Our relation to such thoughts is clarified further in a footnote: A person sees a thing,has an idea, grasps or thinks a thought. When he grasps or thinks a thought he does notcreate it but only comes to stand in a certain relation to what already existed—a differentrelation from seeing a thing or having an idea. (
Ibid
) For Frege the first realm is theuniverse of physical objects, existing in space-time; to it we have access through senseperception. The second realm is our psychological world (or worlds), i.e., eachperson's subjective world of ideas, feelings, and thinking processes; our access here isthrough direct awareness and introspection. Finally, there is a third realm , to becontrasted with the earlier two; it contains thoughts and numbers (maybe more). Thisthird realm is, presumably, analogous to Plato's realm of forms; thus the use of the term'platonism' (in the secondary literature, not by Frege himself).
The Metaphysical Platonist Picture.
Unfortunately, many debates about platonism—andthus also about Frege—remain content with vague, general characterizations of it, such asthat quoted above. In other words, they rely on rather brief descriptions of platonism,mostly in terms of a few metaphors. Typically the following kinds of phrases are used inthese debates: that abstract objects , in particular numbers, really exist , independentlyfrom us , out there ; that they are not just created by us, but discovered ; that themathematician is an explorer , not an inventor ; etc. Such descriptions are usually fol-

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